Page:Calculus Made Easy.pdf/55



have learned how to differentiate simple algebraical functions such as $$x^2+c$$ or $$ax^4$$, and we have now to consider how to tackle the sum of two or more functions.

For instance, let $y=(x^2+c)+(ax^4+b)$; what will its $$\frac {dy}{dx}$$ be? How are we to go to work on this new job?

The answer to this question is quite simple: just differentiate them, one after the other, thus: $\frac {dy}{dx}=2x+4ax^3$. (Ans.)

If you have any doubt whether this is right, try a more general case, working it by first principles. And this is the way.

Let $$y=u+v$$, where u is any function of $$x$$, and $$v$$ any other function of $$x$$. Then, letting $$x$$ increase to $$x+dx$$, $$y$$ will increase to $$y+dy$$; and $$u$$ will increase to $$u+du$$; and $$v$$ to $$v+dv$$.